Q: Trace role in classification of SL(2,R) matrices. I am working on J.Bochi and Avila's article "Uniformly hyperbolic finite-valued $SL(2,\mathbb{R})$-cocycles".As you may know,there is classification of $SL(2,\mathbb{R})$ which depends on the Trace of these matrices.for example if $M$ is our intended matrix, if $\vert Tr M\vert>2$ it is hyperbolic,if$\vert Tr M\vert<2$ then it is elliptic and if $\vert Tr M\vert=2$ then it is parabolic.i want to know why we give such as this names"hyperbolic,elliptic and parabolic" just by traces and why the value 2 is so important to determine the kind of classification.What is the role of trace in this classification ?if you know the answer i would be very happy to give me information or if there exist some free eBook to get some information please let me know.
Thank you in advance.
